NCEA Level 1 Mathematics (90153) 2009 — page 1 of 3
Assessment Schedule – 2009
Mathematics: Use geometric reasoning to solve problems (90153)
Evidence Statement
Question
ONE
(a)
Achievement
Achievement with Merit
c = 122°
Achievement with Excellence
Angle SRP = 98 (co-interior angles // sum to
180)
2c + 116 + 82 + 98 = 540 (interior angles of
pentagon add to 540)
2c = 244°
c = 122°
A = 1 of 1(a) or 1(b)
M = 1 of 1(a) or 1(b) with reasons
E = 1(b) with reasons
Question 1 grade = highest of 1(a) or 1
(b)
(b)
A
A
A
E
B
E
B
C
D
Interior angle pentagon = 108°
Angle BCA = angle BAC = 36°
Reasoning not required.
E
B
C
D
Interior angle pentagon = 108° (Interior angle of
a 5 sided polygon)
BC = BA (regular pentagon)
Therefore ABC is an isosceles triangle.
Angle BCA = angle BAC = 36° (base angles of
an isosceles triangle)
C
D
Interior angle pentagon = 108° (Interior
angle of a 5 sided polygon)
BC = BA (regular pentagon)
Therefore ABC is an isosceles triangle
Angle BCA = angle BAC = 36° (base
angles of an isosceles triangle)
Angle AED + Angle EAC = 108°+72°
= 180° ie co-interior angles.
Since co-interior angles AED and BAC
add up to 180°, AC is parallel to DE
so ACDE is a trapezium.
Or equivalent.
NCEA Level 1 Mathematics (90153) 2009 — page 2 of 3
TWO
(a)
(b)
THREE
(a)
(b)
Angle OCD = 120°
Angle ACD = 39° (alternate angles, parallel
lines)
Angle OCD = 120° (sum of angles on a straight
line add to 180°)
Angle OBC = 51° or
Angle OCB = 51°
Angle BOC = 78° (angle at centre twice angle at
circumference).
Angle OBC = Angle OCB (base angles isos
triangle)
Angle OBC = angle OCB= 51°
a = 21°
Angle GDE = 83 (corresponding angles, parallel
lines
a =180 – 83 – 76 (angle sum of triangle = 180)
therefore a = 21°
Any 2 step angle calculation
Must get to 90-x one of the ways with reasoning.
Any valid proof
Eg: Large triangle ACE:
Angle EAC = 90° and so angle
ACE = 90– x°
Eg: Large triangle ACE:
Angle EAC = 90° (angle between tangent and
radii) and so angle ACE = 90– x°(angle sum of
triangle = 180°)
Eg: Large triangle ACE:
Angle EAC = 90° (angle between
tangent and radii) and so angle ACE =
180 – (90 + x) = 90 – x°(angle sum of
triangle = 180°)
OR Angle ODE = x° and angle
ODB = 90°
OR Angle ODE = x° as angle ODE = angle
OED (isosceles triangle) and angle ODB = 90°
(angle between tangent and radii)
Angle BDC = 180 – (90 + x) = 90 – x° (sum of
angles on a straight line = 180°)
A = 1 of 2(a) or 2(b)
M = 2(a) or 2(b) with reasons
E = 2(b) with reasons
Angle BOC = 78° (angle at centre
twice angle at circumference).
Angle OBC = Angle OCB (base angles
isos triangle)
Angle OBC = angle OCB= 51°
Angle OCA = 51 – x°
Or equivalent.
Question 2 grade = highest of 2(a) or
2(b)
A = 3(a) or 3(b)
M = 3(a) or 3(b) with reasons
E = 3(b) with reasons
Angle ODE = x° as angle ODE = angle
OED (base angles of isosceles triangle
are equal) and angle ODB = 90° (angle
between tangent and radii)
Angle BDC = 180 – (90 + x) = 90 – x°
(sum of angles on a straight line =
180°)
Angle ACE = Angle BDC, therefore
triangle DBC is isosceles.
Question 3 grade = highest of 3(a) or 3
(b)
NCEA Level 1 Mathematics (90153) 2009 — page 3 of 3
Judgement Statement
Achievement
Use geometric reasoning to solve problems
Achievement with Merit
Use, and state, geometric reasons in solving
problems.
Achievement with Excellence
Solve an extended geometrical problem.
2 A from 2 different questions
2 M from 2 different questions
2M+1E
from 3 different questions
OR
2E
Lower case a, m, e may be used throughout the paper to indicate contributing evidence for overall grades for questions.
The circled upper case A, M and E grades shown at the end of each full question are used to make the final judgement.
The following Mathematics-specific marking conventions may also have been used when marking this paper:
 Errors are circled.
 Omissions are indicated by a caret ().
 NS may have been used when there was not sufficient evidence to award a grade.
 CON may have been used to indicate ‘consistency’ where an answer is obtained using a prior, but incorrect answer and NC if the answer is not consistent with
wrong working.
 CAO is used when the ‘correct answer only’ is given and the assessment schedule indicates that more evidence was required.
 # may have been used when a correct answer is obtained but then further (unnecessary) working results in an incorrect final answer being offered.
 RAWW indicates right answer, wrong working.
 R for ‘rounding error’ and PR for ‘premature rounding’ resulting in a significant round-off error in the answer (if the question required evidence for rounding).
 U for incorrect or omitted units (if the question required evidence for units).
 MEI may have been used to indicate where a minor error has been made and ignored.